In this paper, a novel methodology for state estimation of stochastic dynamical systems is proposed. In this formulation, finite-term Karhunen-Loève (KL) expansion is used to approximate the process noise, thus resulting in a non-autonomous deterministic approximation (with parametric uncertainty) of the original stochastic nonlinear system. It is proved that the solutions of the approximate dy...

Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

We will give a summary about the relations between the spectra of the Perron–Frobenius operator and pseudo random sequences for 1-dimensional cases. There are many difficulties to construct general theory of higher-dimensional cases. We will give several examples for these cases.

We describe a computational method of approximating the \physical" or Sinai-Bowen-Ruelle measure of an Anosov system in two dimensions. The approximation may either be viewed as a xed point of an approximate Perron-Frobenius operator or as an invariant measure of a randomly perturbed system.

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

The generalized spectral decomposition of the Frobenius-Perron operator of the tent map with varying height is determined at the band-splitting points. The decomposition includes both decay onto the attracting set and the approach to the asymptotically periodic state on the attractor. Explicit compact expressions for the polynomial eigenstates are obtained using algebraic techniques. (c) 1998 A...

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching aroun...

I illustrate a uniied approach to the study of the decay of correlations in hyperbolic dynamical systems. The decay of correlations or, alternatively, the rate of approach of some initial distribution to an invariant one, is a widely studied problem in dynamical systems as well as in other elds. Although a full understanding is not yet available, some notable results have been obtained in the c...

We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius–Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfunctions is performed.